Download 2013 Prelim Paper

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SENIOR
MATHEMATICS
COMPETITION 2013
Preliminary round Thursday
th
16 May 2013 Time
allowed 1 ½ hours
Instructions Attempt all questions. It is not expected that you will finish them
all. Full working should accompany all solutions. Calculators may be used,
but no other reference material is permitted. Total: 52 marks
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SENIOR MATHEMATICS COMPETITION 2013
1. A function f(x) obeys the rule f(x+1) = f(x) – f(x-1) + 1. Given that f(2) = 8 and f(1) = 3, find f(2013).
2. Prove that √2 is irrational.
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3. For a Sierpinski Triangle (fractal : see diagram below) formed by successively bisecting sides of the shaded
triangles, joining the bisection points and unshading the interior triangle, with S0 having no unshaded triangles,
S1 having one unshaded triangle (and, so, four non-overlapping interior triangles), show that the number of nonoverlapping interior triangles for Sn is given by (3n+1 – 1)/2.
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4. A disease test has a sensitivity of 0.95 (i.e. of those who test positive, 95% have the disease) and a specificity of
0.95 (of those who test negative, 95% do not have the disease).
(a) Show that if the true prevalence of the disease is 40% that over 90% of test results will be correct, and give the
percentages of false positives and false negatives correct to 1d.p.
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(b) Show that if the true prevalence is 2%, that it is less likely that a laboratory positive result is correct, and give the
percentages of false positives and negatives correct to 1d.p.
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(c) Make a generalisation about disease testing.
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5. Let A = sin(π – t) – cos(π – t) and B = sin(π + t) – cos(π + t) . Find an expression for the difference D = A - B in
terms of sin(t) and give the maximum and minimum values this fluctuates between.
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6.
For two numbers x and y, the arithmetic mean is A=(x+y)/2, the geometric mean is G=√(xy) and the harmonic
mean is H=2/(1/x + 1/y). Prove that √(AH/2) - √2 G + G/√2 = 0 for all values of x and y.
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7. Given the function f(n) = 2n + (n-1)(n-2)(n-3)(n-4)x, where n ε N and x ε R, explain why the fifth number in the
sequence 2,4,8,16 … could be any real number, and then determine the value of f(5) if x = -1/24
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8. An equilateral triangle ACB is drawn, with arcs AEC and BDC each of radius AB. Find the ratio
Area of semicircle on AB: Area AECDB.
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9. Rotten teeth: A person (with 32 teeth) is offered the deal of $1million to remove the whole lot, or 1 cent for
the first, 2 cents for the second, 4 cents for the third, 8 cents for the fourth and so on.
(a) Calculate the exact cost for the second option.
(b) If several dentists each extracted fewer than 32 teeth, all using the second cost option, the cost could be as
little as $0.32; but the bus fare from one dentist to the next is a flat rate of $2, giving a maximum cost of $64.32.
Determine the number of dentists and bus rides to give the least cost, and the amount of that cost.
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10. Given that the approximate radius of the earth is 6378km, and assuming (incorrectly) that the earth is
approximately spherical, show that the distance to the horizon from a point at elevation H is given by D(miles) =
1.2√H (feet). Note: 1 mile = 1.62km, 1mile = 5280 feet.
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11. If a*b = a/b + b/a, find an expression for (a-b) * (a+b)
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12. Four aces are face down side by side. Two are chosen at random and turned over. What is the probability
they are different colours, and why?
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13. If f(2x2 – 1) = 2xf(x) for all values of x, and g(t) =[ f(cos(t))]/sin(t), show that
(a) g(t+ π) = g(t) and
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(b) g(2t) = g(t)
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14. For the equation x2 – 12x + k = 0, if there are two rational roots how many possible values of k are there?
Find three of those values and then characterize all the possible values (ie. What features of numbers do you
notice?)
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